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Research Papers

# Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

[+] Author and Article Information
Usama H. Hegazy

Department of Mathematics, Faculty of Science, Al-Azhar University, P.O. Box 1277, Gaza, Palestineu.hejazy@alazhar.edu.ps, uhijazy@yahoo.com

J. Vib. Acoust 132(5), 051004 (Aug 19, 2010) (9 pages) doi:10.1115/1.4001502 History: Received April 11, 2009; Revised January 25, 2010; Published August 19, 2010; Online August 19, 2010

## Abstract

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

###### FIGURES IN THIS ARTICLE
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## Figures

Figure 7

Effect of the in-plane forcing excitation amplitude f1 at Ω1=Ω2=ω1=ω2=9.7520 when (a) f1=150.0, (b) f1=200.0, and (c) f1=250.0. Other parameter values are the same as those in Fig. 6.

Figure 8

Effect of the in-plane forcing excitation amplitude f2 at Ω1=Ω2=ω1=ω2=9.7520 when (a) f2=150.0, (b) f2=200.0, and (c) f2=250.0. Other parameter values are the same as those in Fig. 6.

Figure 1

The model of a rectangular thin plate

Figure 2

Frequency-response curves of case 1 (a1=0, a2≠0). (a) Basic case at F2=300.0, ω2=9.7520, μ=0.4, and β1=5.3841. (b) Effect of excitation force amplitude, (c) effect of natural frequency, (d) effect of damping coefficient, and (e) effect of nonlinear parameter.

Figure 3

Frequency-response curves of case 3 (a1≠0, a2≠0). (a) Basic case at F1=0.5, ω1=9.7520, μ=0.4, α1=4.8782, and α2=7.3725. (b) Effect of excitation force amplitude, (c) effect of natural frequency, (d) effect of damping coefficient, and ((e)–(g)) effect of nonlinear parameters.

Figure 4

Nonresonant time response solution: F1=50.0, f1=0.15, F2=30.0, f2=0.1, Ω1=Ω2=3.2, λ=0.8, ω1=12.0082, ω2=7.8995, μ=0.4, α1=7.9726, β1=3.3766, and α2=β2=7.1790

Figure 5

Simultaneous resonance solution. F1=50.0, f1=0.15, F2=30.0, f2=0.1, λ=1.0253, Ω1=Ω2=ω1=ω2=9.7520, μ=0.4, α1=4.8782, β1=5.3841, and α2=β2=7.3725. Plots of the phase portraits are chosen for final orbits of the simulations.

Figure 6

Numerical solution under various values of the system parameters at simultaneous resonance condition (the solid line is the x-amplitude and the dashed line is the y-amplitude). F1=50.0, f1=0.15, F2=30.0, f2=0.1, λ=1.0253, Ω1=Ω2=ω1=ω2=9.7520, μ=0.4, α1=4.8782, β1=5.3841, and α2=β2=7.3725. ((a) and (b)) Effects of transverse excitation amplitudes, (c) effect of damping coefficient, and ((d)–(g)) effects of nonlinear parameters.

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